We present the results of the first, deep Atacama Large Millimeter Array (ALMA) imaging covering the full ≃4.5 arcmin2 of the Hubble Ultra Deep Field (HUDF) imaged with Wide Field Camera 3/IR on HST. Using a 45-pointing mosaic, we have obtained a homogeneous 1.3-mm image reaching σ1.3 ≃ 35 μJy, at a resolution of ≃0.7 arcsec. From an initial list of ≃50 > 3.5σ peaks, a rigorous analysis confirms 16 sources with S1.3 > 120 μJy. All of these have secure galaxy counterparts with robust redshifts (〈z〉 = 2.15). Due to the unparalleled supporting data, the physical properties of the ALMA sources are well constrained, including their stellar masses (M*) and UV+FIR star formation rates (SFR). Our results show that stellar mass is the best predictor of SFR in the high-redshift Universe; indeed at z ≥ 2 our ALMA sample contains seven of the nine galaxies in the HUDF with M* ≥ 2 × 1010 M⊙, and we detect only one galaxy at z > 3.5, reflecting the rapid drop-off of high-mass galaxies with increasing redshift. The detections, coupled with stacking, allow us to probe the redshift/mass distribution of the 1.3-mm background down to S1.3 ≃ 10 μJy. We find strong evidence for a steep star-forming ‘main sequence’ at z ≃ 2, with SFR ∝M* and a mean specific SFR ≃ 2.2 Gyr−1. Moreover, we find that ≃85 per cent of total star formation at z ≃ 2 is enshrouded in dust, with ≃65 per cent of all star formation at this epoch occurring in high-mass galaxies (M* > 2 × 1010 M⊙), for which the average obscured:unobscured SF ratio is ≃200. Finally, we revisit the cosmic evolution of SFR density; we find this peaks at z ≃ 2.5, and that the star-forming Universe transits from primarily unobscured to primarily obscured at z ≃ 4.